The dimension of a vector space, the trace of a linear map and the character of a group representation can be seen as instances of an abstract notion of character. I will explain how loop spaces and fixed points provide nonlinear analogues of these notions in the brave new world of derived algebraic geometry, a strange blend of topology and geometry. Following ideas from topological field theory, we'll see how passing back to the linear world results in interesting trace and character formulas. This is joint work with David Nadler (Berkeley).